\(\int \cot ^7(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x)) \, dx\) [675]

Optimal result

Integrand size = 27, antiderivative size = 113 \[ \int \cot ^7(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \cot ^8(c+d x)}{8 d}-\frac {a \cot ^{10}(c+d x)}{5 d}-\frac {a \cot ^{12}(c+d x)}{12 d}+\frac {a \csc ^7(c+d x)}{7 d}-\frac {a \csc ^9(c+d x)}{3 d}+\frac {3 a \csc ^{11}(c+d x)}{11 d}-\frac {a \csc ^{13}(c+d x)}{13 d} \] Output:

-1/8*a*cot(d*x+c)^8/d-1/5*a*cot(d*x+c)^10/d-1/12*a*cot(d*x+c)^12/d+1/7*a*c 
sc(d*x+c)^7/d-1/3*a*csc(d*x+c)^9/d+3/11*a*csc(d*x+c)^11/d-1/13*a*csc(d*x+c 
)^13/d
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.14 \[ \int \cot ^7(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \csc ^6(c+d x)}{6 d}+\frac {a \csc ^7(c+d x)}{7 d}-\frac {3 a \csc ^8(c+d x)}{8 d}-\frac {a \csc ^9(c+d x)}{3 d}+\frac {3 a \csc ^{10}(c+d x)}{10 d}+\frac {3 a \csc ^{11}(c+d x)}{11 d}-\frac {a \csc ^{12}(c+d x)}{12 d}-\frac {a \csc ^{13}(c+d x)}{13 d} \] Input:

Integrate[Cot[c + d*x]^7*Csc[c + d*x]^7*(a + a*Sin[c + d*x]),x]
 

Output:

(a*Csc[c + d*x]^6)/(6*d) + (a*Csc[c + d*x]^7)/(7*d) - (3*a*Csc[c + d*x]^8) 
/(8*d) - (a*Csc[c + d*x]^9)/(3*d) + (3*a*Csc[c + d*x]^10)/(10*d) + (3*a*Cs 
c[c + d*x]^11)/(11*d) - (a*Csc[c + d*x]^12)/(12*d) - (a*Csc[c + d*x]^13)/( 
13*d)
 

Rubi [A] (warning: unable to verify)

Time = 0.51 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.89, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {3042, 3313, 3042, 25, 3086, 25, 244, 2009, 3087, 243, 49, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \cot ^7(c+d x) \csc ^7(c+d x) (a \sin (c+d x)+a) \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\cos (c+d x)^7 (a \sin (c+d x)+a)}{\sin (c+d x)^{14}}dx\)

\(\Big \downarrow \) 3313

\(\displaystyle a \int \cot ^7(c+d x) \csc ^7(c+d x)dx+a \int \cot ^7(c+d x) \csc ^6(c+d x)dx\)

\(\Big \downarrow \) 3042

\(\displaystyle a \int -\sec \left (c+d x-\frac {\pi }{2}\right )^6 \tan \left (c+d x-\frac {\pi }{2}\right )^7dx+a \int -\sec \left (c+d x-\frac {\pi }{2}\right )^7 \tan \left (c+d x-\frac {\pi }{2}\right )^7dx\)

\(\Big \downarrow \) 25

\(\displaystyle -a \int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^6 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^7dx-a \int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^7 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^7dx\)

\(\Big \downarrow \) 3086

\(\displaystyle -\frac {a \int -\csc ^6(c+d x) \left (1-\csc ^2(c+d x)\right )^3d\csc (c+d x)}{d}-a \int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^6 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^7dx\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {a \int \csc ^6(c+d x) \left (1-\csc ^2(c+d x)\right )^3d\csc (c+d x)}{d}-a \int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^6 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^7dx\)

\(\Big \downarrow \) 244

\(\displaystyle \frac {a \int \left (-\csc ^{12}(c+d x)+3 \csc ^{10}(c+d x)-3 \csc ^8(c+d x)+\csc ^6(c+d x)\right )d\csc (c+d x)}{d}-a \int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^6 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^7dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -a \int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^6 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^7dx-\frac {a \left (\frac {1}{13} \csc ^{13}(c+d x)-\frac {3}{11} \csc ^{11}(c+d x)+\frac {1}{3} \csc ^9(c+d x)-\frac {1}{7} \csc ^7(c+d x)\right )}{d}\)

\(\Big \downarrow \) 3087

\(\displaystyle -\frac {a \int -\cot ^7(c+d x) \left (\cot ^2(c+d x)+1\right )^2d(-\cot (c+d x))}{d}-\frac {a \left (\frac {1}{13} \csc ^{13}(c+d x)-\frac {3}{11} \csc ^{11}(c+d x)+\frac {1}{3} \csc ^9(c+d x)-\frac {1}{7} \csc ^7(c+d x)\right )}{d}\)

\(\Big \downarrow \) 243

\(\displaystyle -\frac {a \int -\cot ^3(c+d x) \left (\cot ^2(c+d x)+1\right )^2d\cot ^2(c+d x)}{2 d}-\frac {a \left (\frac {1}{13} \csc ^{13}(c+d x)-\frac {3}{11} \csc ^{11}(c+d x)+\frac {1}{3} \csc ^9(c+d x)-\frac {1}{7} \csc ^7(c+d x)\right )}{d}\)

\(\Big \downarrow \) 49

\(\displaystyle -\frac {a \int \left (-\cot ^5(c+d x)+2 \cot ^4(c+d x)-\cot ^3(c+d x)\right )d\cot ^2(c+d x)}{2 d}-\frac {a \left (\frac {1}{13} \csc ^{13}(c+d x)-\frac {3}{11} \csc ^{11}(c+d x)+\frac {1}{3} \csc ^9(c+d x)-\frac {1}{7} \csc ^7(c+d x)\right )}{d}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {a \left (\frac {1}{6} \cot ^6(c+d x)-\frac {2}{5} \cot ^5(c+d x)+\frac {1}{4} \cot ^4(c+d x)\right )}{2 d}-\frac {a \left (\frac {1}{13} \csc ^{13}(c+d x)-\frac {3}{11} \csc ^{11}(c+d x)+\frac {1}{3} \csc ^9(c+d x)-\frac {1}{7} \csc ^7(c+d x)\right )}{d}\)

Input:

Int[Cot[c + d*x]^7*Csc[c + d*x]^7*(a + a*Sin[c + d*x]),x]
 

Output:

-1/2*(a*(Cot[c + d*x]^4/4 - (2*Cot[c + d*x]^5)/5 + Cot[c + d*x]^6/6))/d - 
(a*(-1/7*Csc[c + d*x]^7 + Csc[c + d*x]^9/3 - (3*Csc[c + d*x]^11)/11 + Csc[ 
c + d*x]^13/13))/d
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 49

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int 
[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] 
&& IGtQ[m, 0] && IGtQ[m + n + 2, 0]
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 244

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand 
Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p 
, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3086

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( 
n_.), x_Symbol] :> Simp[a/f   Subst[Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2 
), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2 
] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])
 

rule 3087

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_S 
ymbol] :> Simp[1/f   Subst[Int[(b*x)^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + 
f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n - 1) 
/2] && LtQ[0, n, m - 1])
 

rule 3313

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_ 
) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[a   Int[Cos[e + f*x]^ 
p*(d*Sin[e + f*x])^n, x], x] + Simp[b/d   Int[Cos[e + f*x]^p*(d*Sin[e + f*x 
])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n, p}, x] && IntegerQ[(p - 1)/2 
] && IntegerQ[n] && ((LtQ[p, 0] && NeQ[a^2 - b^2, 0]) || LtQ[0, n, p - 1] | 
| LtQ[p + 1, -n, 2*p + 1])
 

Maple [A] (verified)

Time = 0.80 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.78

method	result	size
derivativedivides	\(-\frac {a \left (\frac {\csc \left (d x +c \right )^{13}}{13}+\frac {\csc \left (d x +c \right )^{12}}{12}-\frac {3 \csc \left (d x +c \right )^{11}}{11}-\frac {3 \csc \left (d x +c \right )^{10}}{10}+\frac {\csc \left (d x +c \right )^{9}}{3}+\frac {3 \csc \left (d x +c \right )^{8}}{8}-\frac {\csc \left (d x +c \right )^{7}}{7}-\frac {\csc \left (d x +c \right )^{6}}{6}\right )}{d}\)	\(88\)
default	\(-\frac {a \left (\frac {\csc \left (d x +c \right )^{13}}{13}+\frac {\csc \left (d x +c \right )^{12}}{12}-\frac {3 \csc \left (d x +c \right )^{11}}{11}-\frac {3 \csc \left (d x +c \right )^{10}}{10}+\frac {\csc \left (d x +c \right )^{9}}{3}+\frac {3 \csc \left (d x +c \right )^{8}}{8}-\frac {\csc \left (d x +c \right )^{7}}{7}-\frac {\csc \left (d x +c \right )^{6}}{6}\right )}{d}\)	\(88\)
risch	\(-\frac {32 a \left (8580 i {\mathrm e}^{19 i \left (d x +c \right )}+5005 \,{\mathrm e}^{20 i \left (d x +c \right )}+28600 i {\mathrm e}^{17 i \left (d x +c \right )}+10010 \,{\mathrm e}^{18 i \left (d x +c \right )}+70460 i {\mathrm e}^{15 i \left (d x +c \right )}+24024 \,{\mathrm e}^{16 i \left (d x +c \right )}+80400 i {\mathrm e}^{13 i \left (d x +c \right )}+3003 \,{\mathrm e}^{14 i \left (d x +c \right )}+70460 i {\mathrm e}^{11 i \left (d x +c \right )}-3003 \,{\mathrm e}^{12 i \left (d x +c \right )}+28600 i {\mathrm e}^{9 i \left (d x +c \right )}-24024 \,{\mathrm e}^{10 i \left (d x +c \right )}+8580 i {\mathrm e}^{7 i \left (d x +c \right )}-10010 \,{\mathrm e}^{8 i \left (d x +c \right )}-5005 \,{\mathrm e}^{6 i \left (d x +c \right )}\right )}{15015 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{13}}\)	\(193\)

Input:

int(cot(d*x+c)^7*csc(d*x+c)^7*(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
 

Output:

-a/d*(1/13*csc(d*x+c)^13+1/12*csc(d*x+c)^12-3/11*csc(d*x+c)^11-3/10*csc(d* 
x+c)^10+1/3*csc(d*x+c)^9+3/8*csc(d*x+c)^8-1/7*csc(d*x+c)^7-1/6*csc(d*x+c)^ 
6)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.42 \[ \int \cot ^7(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {17160 \, a \cos \left (d x + c\right )^{6} - 11440 \, a \cos \left (d x + c\right )^{4} + 4160 \, a \cos \left (d x + c\right )^{2} + 1001 \, {\left (20 \, a \cos \left (d x + c\right )^{6} - 15 \, a \cos \left (d x + c\right )^{4} + 6 \, a \cos \left (d x + c\right )^{2} - a\right )} \sin \left (d x + c\right ) - 640 \, a}{120120 \, {\left (d \cos \left (d x + c\right )^{12} - 6 \, d \cos \left (d x + c\right )^{10} + 15 \, d \cos \left (d x + c\right )^{8} - 20 \, d \cos \left (d x + c\right )^{6} + 15 \, d \cos \left (d x + c\right )^{4} - 6 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \] Input:

integrate(cot(d*x+c)^7*csc(d*x+c)^7*(a+a*sin(d*x+c)),x, algorithm="fricas" 
)
 

Output:

-1/120120*(17160*a*cos(d*x + c)^6 - 11440*a*cos(d*x + c)^4 + 4160*a*cos(d* 
x + c)^2 + 1001*(20*a*cos(d*x + c)^6 - 15*a*cos(d*x + c)^4 + 6*a*cos(d*x + 
 c)^2 - a)*sin(d*x + c) - 640*a)/((d*cos(d*x + c)^12 - 6*d*cos(d*x + c)^10 
 + 15*d*cos(d*x + c)^8 - 20*d*cos(d*x + c)^6 + 15*d*cos(d*x + c)^4 - 6*d*c 
os(d*x + c)^2 + d)*sin(d*x + c))
 

Sympy [F(-1)]

Timed out. \[ \int \cot ^7(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x)) \, dx=\text {Timed out} \] Input:

integrate(cot(d*x+c)**7*csc(d*x+c)**7*(a+a*sin(d*x+c)),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.81 \[ \int \cot ^7(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x)) \, dx=\frac {20020 \, a \sin \left (d x + c\right )^{7} + 17160 \, a \sin \left (d x + c\right )^{6} - 45045 \, a \sin \left (d x + c\right )^{5} - 40040 \, a \sin \left (d x + c\right )^{4} + 36036 \, a \sin \left (d x + c\right )^{3} + 32760 \, a \sin \left (d x + c\right )^{2} - 10010 \, a \sin \left (d x + c\right ) - 9240 \, a}{120120 \, d \sin \left (d x + c\right )^{13}} \] Input:

integrate(cot(d*x+c)^7*csc(d*x+c)^7*(a+a*sin(d*x+c)),x, algorithm="maxima" 
)
 

Output:

1/120120*(20020*a*sin(d*x + c)^7 + 17160*a*sin(d*x + c)^6 - 45045*a*sin(d* 
x + c)^5 - 40040*a*sin(d*x + c)^4 + 36036*a*sin(d*x + c)^3 + 32760*a*sin(d 
*x + c)^2 - 10010*a*sin(d*x + c) - 9240*a)/(d*sin(d*x + c)^13)
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.81 \[ \int \cot ^7(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x)) \, dx=\frac {20020 \, a \sin \left (d x + c\right )^{7} + 17160 \, a \sin \left (d x + c\right )^{6} - 45045 \, a \sin \left (d x + c\right )^{5} - 40040 \, a \sin \left (d x + c\right )^{4} + 36036 \, a \sin \left (d x + c\right )^{3} + 32760 \, a \sin \left (d x + c\right )^{2} - 10010 \, a \sin \left (d x + c\right ) - 9240 \, a}{120120 \, d \sin \left (d x + c\right )^{13}} \] Input:

integrate(cot(d*x+c)^7*csc(d*x+c)^7*(a+a*sin(d*x+c)),x, algorithm="giac")
 

Output:

1/120120*(20020*a*sin(d*x + c)^7 + 17160*a*sin(d*x + c)^6 - 45045*a*sin(d* 
x + c)^5 - 40040*a*sin(d*x + c)^4 + 36036*a*sin(d*x + c)^3 + 32760*a*sin(d 
*x + c)^2 - 10010*a*sin(d*x + c) - 9240*a)/(d*sin(d*x + c)^13)
 

Mupad [B] (verification not implemented)

Time = 32.11 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.81 \[ \int \cot ^7(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {-\frac {a\,{\sin \left (c+d\,x\right )}^7}{6}-\frac {a\,{\sin \left (c+d\,x\right )}^6}{7}+\frac {3\,a\,{\sin \left (c+d\,x\right )}^5}{8}+\frac {a\,{\sin \left (c+d\,x\right )}^4}{3}-\frac {3\,a\,{\sin \left (c+d\,x\right )}^3}{10}-\frac {3\,a\,{\sin \left (c+d\,x\right )}^2}{11}+\frac {a\,\sin \left (c+d\,x\right )}{12}+\frac {a}{13}}{d\,{\sin \left (c+d\,x\right )}^{13}} \] Input:

int((cot(c + d*x)^7*(a + a*sin(c + d*x)))/sin(c + d*x)^7,x)
 

Output:

-(a/13 + (a*sin(c + d*x))/12 - (3*a*sin(c + d*x)^2)/11 - (3*a*sin(c + d*x) 
^3)/10 + (a*sin(c + d*x)^4)/3 + (3*a*sin(c + d*x)^5)/8 - (a*sin(c + d*x)^6 
)/7 - (a*sin(c + d*x)^7)/6)/(d*sin(c + d*x)^13)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 312, normalized size of antiderivative = 2.76 \[ \int \cot ^7(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \left (-9240 \cos \left (d x +c \right ) \cot \left (d x +c \right )^{5} \csc \left (d x +c \right )^{7} \sin \left (d x +c \right )^{6}+10752 \cos \left (d x +c \right ) \cot \left (d x +c \right )^{3} \csc \left (d x +c \right )^{7} \sin \left (d x +c \right )^{6}-7588 \cos \left (d x +c \right ) \cot \left (d x +c \right ) \csc \left (d x +c \right )^{7} \sin \left (d x +c \right )^{6}-110880 \cot \left (d x +c \right )^{6} \csc \left (d x +c \right )^{7} \sin \left (d x +c \right )^{7}-110880 \cot \left (d x +c \right )^{6} \csc \left (d x +c \right )^{7} \sin \left (d x +c \right )^{6}+61320 \cot \left (d x +c \right )^{4} \csc \left (d x +c \right )^{7} \sin \left (d x +c \right )^{7}+60480 \cot \left (d x +c \right )^{4} \csc \left (d x +c \right )^{7} \sin \left (d x +c \right )^{6}-28448 \cot \left (d x +c \right )^{2} \csc \left (d x +c \right )^{7} \sin \left (d x +c \right )^{7}-26880 \cot \left (d x +c \right )^{2} \csc \left (d x +c \right )^{7} \sin \left (d x +c \right )^{6}-7588 \csc \left (d x +c \right )^{7} \sin \left (d x +c \right )^{7}+7680 \csc \left (d x +c \right )^{7} \sin \left (d x +c \right )^{6}-6125 \sin \left (d x +c \right )^{6}+19600\right )}{1441440 \sin \left (d x +c \right )^{6} d} \] Input:

int(cot(d*x+c)^7*csc(d*x+c)^7*(a+a*sin(d*x+c)),x)
 

Output:

(a*( - 9240*cos(c + d*x)*cot(c + d*x)**5*csc(c + d*x)**7*sin(c + d*x)**6 + 
 10752*cos(c + d*x)*cot(c + d*x)**3*csc(c + d*x)**7*sin(c + d*x)**6 - 7588 
*cos(c + d*x)*cot(c + d*x)*csc(c + d*x)**7*sin(c + d*x)**6 - 110880*cot(c 
+ d*x)**6*csc(c + d*x)**7*sin(c + d*x)**7 - 110880*cot(c + d*x)**6*csc(c + 
 d*x)**7*sin(c + d*x)**6 + 61320*cot(c + d*x)**4*csc(c + d*x)**7*sin(c + d 
*x)**7 + 60480*cot(c + d*x)**4*csc(c + d*x)**7*sin(c + d*x)**6 - 28448*cot 
(c + d*x)**2*csc(c + d*x)**7*sin(c + d*x)**7 - 26880*cot(c + d*x)**2*csc(c 
 + d*x)**7*sin(c + d*x)**6 - 7588*csc(c + d*x)**7*sin(c + d*x)**7 + 7680*c 
sc(c + d*x)**7*sin(c + d*x)**6 - 6125*sin(c + d*x)**6 + 19600))/(1441440*s 
in(c + d*x)**6*d)